Derived induction theory for the K-theory of modular group algebras

TL;DR

This work proves an induction theorem for the higher algebraic K-theory of group algebras $kG$ over finite fields of characteristic $p$, by introducing $p$-isolated groups that permit reduction to $p$-subgroups. The core construction passes through a Cartan cofiber sequence of spectral Mackey functors, showing that the singular K-theory $\\K^{\sg}_G(k)$ carries a spectral Green functor structure and that induction can be analyzed via the derived induction theory of Mathew--Naumann--Noel. A key step is proving that $\\K^{\sg}_G(k)$ controls the $p$-local part of $\\K_*(kG)$, enabling a colimit decomposition over the orbit category $\,\mathcal{O}_p(G)$; under a trivial intersection condition for Sylow $p$-subgroups this simplifies to Weyl-coinvariants. The results yield explicit computations for many families, including $p$-isolated groups with $|S|=p$ (e.g., dihedral and symmetric groups in suitable ranges), and provide a framework for further calculations with larger Sylow subgroups. Overall, the paper extends TC-based insights by delivering a concrete K-theory induction mechanism that enlarges the class of groups with computable $p$-adic K-groups.

Abstract

We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces calculations to calculations for their $p$-subgroups. We do so by showing that the stable module categories of $kH$ as $H$ ranges over subgroups of $G$ assemble into a categorical Green functor, which results in a spectral Green functor structure on K-theory. By general induction theory, this reduces proving a spectrum-level induction statement to proving an induction statement on $π_0$ Green functors, which we accomplish using modular representation theory. For $p$-isolated groups with Sylow $p$-subgroups of order $p$, we produce explicit new calculations of K-groups.

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2.1.1
• Proposition 2.1.4
• proof
• Proposition 2.1.5
• proof
• Proposition 2.1.6